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An Implementable Splitting Algorithm for the \(\ell _1\)-norm Regularized Split Feasibility Problem

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Abstract

The split feasibility problem (SFP) captures a wide range of inverse problems, such as signal processing, image reconstruction, and so on. Recently, applications of \(\ell _1\)-norm regularization to linear inverse problems, a special case of SFP, have been received a considerable amount of attention in the signal/image processing and statistical learning communities. However, the study of the \(\ell _1\)-norm regularized SFP still deserves attention, especially in terms of algorithmic issues. In this paper, we shall propose an algorithm for solving the \(\ell _1\)-norm regularized SFP. More specifically, we first formulate the \(\ell _1\)-norm regularized SFP as a separable convex minimization problem with linear constraints, and then introduce our splitting method, which takes advantage of the separable structure and gives rise to subproblems with closed-form solutions. We prove global convergence of the proposed algorithm under certain mild conditions. Moreover, numerical experiments on an image deblurring problem verify the efficiency of our algorithm.

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Acknowledgments

The authors would like to thank the two anonymous referees for their constructive comments, which significantly improved the presentation of this paper. The first two authors were supported by National Natural Science Foundation of China (11301123; 11171083) and the Zhejiang Provincial NSFC Grant No. LZ14A010003, and the third author in part by NSC 102-2115-M-110-001-MY3.

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Authors and Affiliations

  1. Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou, 310018, Zhejiang, China

    Hongjin He, Chen Ling & Hong-Kun Xu

  2. Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, 80424, Taiwan

    Hong-Kun Xu

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  1. Hongjin He
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  2. Chen Ling
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Correspondence to Hong-Kun Xu.

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He, H., Ling, C. & Xu, HK. An Implementable Splitting Algorithm for the \(\ell _1\)-norm Regularized Split Feasibility Problem. J Sci Comput 67, 281–298 (2016). https://doi.org/10.1007/s10915-015-0078-4

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